Relationships within triangles Geometry Unit 5
Description: Relationships within triangles Geometry Unit 5 Midsegments of a triangle What is a midsegment? A midsegment connects the midpoints of two segments of a triangle. Given diagram below: Construction of a midpoint
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slide1. Relationships within triangles Geometry Unit 5<br>
slide2. Midsegments of a triangle<br>
slide3. What is a midsegment? A midsegment connects the midpoints of two segments of a triangle.
Given diagram below:<br>
slide4. Construction of a midpoint http://www.mathsisfun.com/geometry/constructions.html<br>
slide5. Observations What do you notice about the midsegments you created?
Lengths?
Lines?
Parallels?
Perpendiculars?
Intersections?<br>
slide6. Properties of midsegements Midsegment Theorem:
The segment that joins the midpoints of a pair of sides of a triangle is:
Parallel to the third side
Half as long as the third side<br>
slide8. Usefulness When would you use the midpoints of a triangle?
When trying to determine relationships within a triangle.
When trying to solve for a variable or a length
Application problems when needing to cut across the center and find the distance.<br>
slide9. Perpendicular bisectors in a triangle<br>
slide10. What is a perpendicular Bisector? Perpendicular bisectors:
Divide the line segment into two congruent parts
Intersects the line at a right angle<br>
slide11. Construction of a Perpendicular Bisector http://www.mathsisfun.com/geometry/constructions.html<br>
slide12. Perpendicular bisectors What do you notice about the perpendicular bisectors of the triangles?
Lengths?
Lines?
Parallels?
Perpendiculars?
Intersections?<br>
slide13. Practice On triangles three and four on your worksheet create the perpendicular bisectors for the triangles.<br>
slide14. Circumcenter The point of concurrency of the three perpendicular bisectors of the sides of a triangle.
This point will be equidistant from each vertex.<br>
slide15. Circumcenter What will the circumcenter look like in the following triangles?
Acute?
Right?
Obtuse?
How do you know? Do these findings make sense?<br>
slide16. Angle bisectors in triangles<br>
slide17. What Is an angle bisector? An angle bisector is a ray or segment which cuts an angle into two congruent angles.<br>
slide18. Construction of an angle bisector http://www.mathsisfun.com/geometry/constructions.html<br>
slide19. Observations What do you notice about the angle bisectors of the triangles?
Lengths?
Lines?
Parallels?
Perpendiculars?
Intersections?<br>
slide20. incenter The incenter is the point of concurrency of all three angle bisectors of a triangle.
The incenter is equidistant from all the edges of the triangle.<br>
slide21. incenter What will the incenter look like in each of the following triangles?
Acute?
Right?
Obtuse?
How do you know?<br>
slide22. Medians in triangles<br>
slide23. What is a median? A median of a triangle is the line segment which joins a vertex to the midpoint of the opposite side.<br>
slide24. Construction of a Median How could you use your compass and protractor to create a median of a triangle?
Lets practice on one of the triangles on your paper.<br>
slide25. Observations Do the three medians meet in a point?
YES! We call this the centroid of the triangle.
What do we notice about the centroid of the triangle?
Lengths?
Angles?
Segments?<br>
slide26. Concurrency of medians theorem States the medians of a triangle will intersect in a point that is 2/3 the distance from the vertices to the midpoint of the opposite side.<br>
slide27. Concurrency of medians of a triangle<br>
slide28. Altitudes in triangles<br>
slide29. What is an altitude? The line segment from a vertex perpendicular to the opposite side.<br>
slide30. Construction of an Altitude Can I construct the altitude of a triangle?
How!!!!!!?????
http://www.mathsisfun.com/geometry/construct-perpnotline.html<br>
slide31. Lets try!!! Lets create the altitudes of the three angles in one of the triangles on your page.
What do you notice about the altitudes?<br>
slide32. Orthocenter The point of concurrency of the altitudes of a triangle is called the orthocenter.
What do you notice about the orthocenter of the following triangles:
Acute
Obtuse
Right<br>
slide33. Observations<br>
slide34. Constructions On your triangle page, find the following for a single triangle. You may want to erase your marks after each.
Circumcenter
Incenter
Centroid
Orthocenter<br>
slide35. Group Work After you have completed each of the constructions and have them clearly labeled.
Note all the similarities you can about these<br>
slide36. Similarities Acute triangle- all points are inside the triangle
Obtuse triangle- all points are outside the triangle
Right triangle- orthocenter on vertex of right angle, circumcenter is on the midpoint of the hypotenuse<br>
slide37. Similarities For the general case of a triangle:
The orthocenter, circumcenter, and centroid are always collinear
The distance from the orthocenter to the centroid is twice the distance from the centroid to the circumcenter.<br>
slide38. Review Perpendicular bisectors meet at the circumcenter.
Angle bisectors meet at the incenter.
Medians meet at the centroid.
Altitudes meet at the orthocenter.<br>
slide39. Inequalities in triangles<br>
slide40. Inequalities in two triangles<br>
slide2. Midsegments of a triangle<br>
slide3. What is a midsegment? A midsegment connects the midpoints of two segments of a triangle.
Given diagram below:<br>
slide4. Construction of a midpoint http://www.mathsisfun.com/geometry/constructions.html<br>
slide5. Observations What do you notice about the midsegments you created?
Lengths?
Lines?
Parallels?
Perpendiculars?
Intersections?<br>
slide6. Properties of midsegements Midsegment Theorem:
The segment that joins the midpoints of a pair of sides of a triangle is:
Parallel to the third side
Half as long as the third side<br>
slide8. Usefulness When would you use the midpoints of a triangle?
When trying to determine relationships within a triangle.
When trying to solve for a variable or a length
Application problems when needing to cut across the center and find the distance.<br>
slide9. Perpendicular bisectors in a triangle<br>
slide10. What is a perpendicular Bisector? Perpendicular bisectors:
Divide the line segment into two congruent parts
Intersects the line at a right angle<br>
slide11. Construction of a Perpendicular Bisector http://www.mathsisfun.com/geometry/constructions.html<br>
slide12. Perpendicular bisectors What do you notice about the perpendicular bisectors of the triangles?
Lengths?
Lines?
Parallels?
Perpendiculars?
Intersections?<br>
slide13. Practice On triangles three and four on your worksheet create the perpendicular bisectors for the triangles.<br>
slide14. Circumcenter The point of concurrency of the three perpendicular bisectors of the sides of a triangle.
This point will be equidistant from each vertex.<br>
slide15. Circumcenter What will the circumcenter look like in the following triangles?
Acute?
Right?
Obtuse?
How do you know? Do these findings make sense?<br>
slide16. Angle bisectors in triangles<br>
slide17. What Is an angle bisector? An angle bisector is a ray or segment which cuts an angle into two congruent angles.<br>
slide18. Construction of an angle bisector http://www.mathsisfun.com/geometry/constructions.html<br>
slide19. Observations What do you notice about the angle bisectors of the triangles?
Lengths?
Lines?
Parallels?
Perpendiculars?
Intersections?<br>
slide20. incenter The incenter is the point of concurrency of all three angle bisectors of a triangle.
The incenter is equidistant from all the edges of the triangle.<br>
slide21. incenter What will the incenter look like in each of the following triangles?
Acute?
Right?
Obtuse?
How do you know?<br>
slide22. Medians in triangles<br>
slide23. What is a median? A median of a triangle is the line segment which joins a vertex to the midpoint of the opposite side.<br>
slide24. Construction of a Median How could you use your compass and protractor to create a median of a triangle?
Lets practice on one of the triangles on your paper.<br>
slide25. Observations Do the three medians meet in a point?
YES! We call this the centroid of the triangle.
What do we notice about the centroid of the triangle?
Lengths?
Angles?
Segments?<br>
slide26. Concurrency of medians theorem States the medians of a triangle will intersect in a point that is 2/3 the distance from the vertices to the midpoint of the opposite side.<br>
slide27. Concurrency of medians of a triangle<br>
slide28. Altitudes in triangles<br>
slide29. What is an altitude? The line segment from a vertex perpendicular to the opposite side.<br>
slide30. Construction of an Altitude Can I construct the altitude of a triangle?
How!!!!!!?????
http://www.mathsisfun.com/geometry/construct-perpnotline.html<br>
slide31. Lets try!!! Lets create the altitudes of the three angles in one of the triangles on your page.
What do you notice about the altitudes?<br>
slide32. Orthocenter The point of concurrency of the altitudes of a triangle is called the orthocenter.
What do you notice about the orthocenter of the following triangles:
Acute
Obtuse
Right<br>
slide33. Observations<br>
slide34. Constructions On your triangle page, find the following for a single triangle. You may want to erase your marks after each.
Circumcenter
Incenter
Centroid
Orthocenter<br>
slide35. Group Work After you have completed each of the constructions and have them clearly labeled.
Note all the similarities you can about these<br>
slide36. Similarities Acute triangle- all points are inside the triangle
Obtuse triangle- all points are outside the triangle
Right triangle- orthocenter on vertex of right angle, circumcenter is on the midpoint of the hypotenuse<br>
slide37. Similarities For the general case of a triangle:
The orthocenter, circumcenter, and centroid are always collinear
The distance from the orthocenter to the centroid is twice the distance from the centroid to the circumcenter.<br>
slide38. Review Perpendicular bisectors meet at the circumcenter.
Angle bisectors meet at the incenter.
Medians meet at the centroid.
Altitudes meet at the orthocenter.<br>
slide39. Inequalities in triangles<br>
slide40. Inequalities in two triangles<br>